Toy Models of Interconnected Networks ofResources and Consumers

Manish Nag1, Cesar Flores2, and Susanne Kortsch3

1Department of Sociology, Princeton University2School of Physics, Georgia Institute of Technology

3Faculty of Biosciences, Fisheries and Economics, Norwegian College ofFishery Science

16 September 2013

Abstract

Real networks are not isolated, but are interlinked with and dependent on otherreal networks. This paper presents a set of tools for studying multiple interconnectednetworks. Utilizing simulation, a controlled environment can be created in which to un-derstand multinetwork dynamics at small and large scales. The paper uses simulationto look at interconnected networks of resources and consumers. The paper examineswhen the removal of any single node in the resource network has the largest impact onthe consumer network. The paper finds that it’s import to focus on the network layerthat connects the the separate networks of resources and consumers. In a controlledsetting, the paper finds that increasing network density of the resource-consumer layerhas a positive impact on minimizing negative consumer impact. Increasing Shannon’sdiversity index on outgoing degree in the resource-consumer layer minimizes worst casenegative impact. Increasing Shannon’s diversity on indegree maximizes negative im-pact. The paper also finds that separate networks are more ”attuned” to one anotherunder conditions of higher utilization. There is a wide avenue of future opportuni-ties for further research in this new field of burgeoning interest in the social networksliterature.

1 Introduction

Real networks are not isolated, but are interlinked with and dependent on other realnetworks (Radicchi and Arenas, 2013, Buldyrev et al., 2010). Examples of coupled net-works are: the communication and power grid system, various transportation systems

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and energy supply networks. Analyses show that results from coupled networks differfrom results based on single networks (Buldyrev et al., 2010). While studies performedon isolated scale-free networks have been shown to be robust against random failuresof nodes and edges, the same is not true for interdependent scale-free networks that aremore vulnerable to similar failures (Radicchi and Arenas, 2013). Failures in nodes andedges of one network cascade to the dependent networks, and as the functionality ofthe infrastructure depends on the functionality on the other networks, such failures canlead to a breakdown of the whole systems. Therefore studying the interdependenciesof networks is important, because it may enable us to understand real networks betterand to make their infrastructures more efficient and robust to failures.

Most real networks are scale-free (e.g. internet, WWW, social networks, aviationnetwork, and networks in biology) which mean they follow a power-law form in theirdegree distribution (Gao et al., 2011). An example of a scale-free network is the airtransportation network. Due to the scale-free property, the aviation network may bevulnerable to targeted attacks, but less vulnerable to random attacks. The aviationnetwork, which is a consumer network, is depended on the functionality of the resourcenetwork, which is the oil supply network that fuels the aircrafts.

In the present study, we designed a toy-model (simplified model of real networks)inspired by the interdependence between a consumer (aviation) and a resource (oil)network. For example, A Boeing 747 uses 5 gallons of fuel per mile (www.Boeing.com),which corresponds to 8.4 miles travelled per barrel of oil. In the case of the aviationand the oil network, one can imagine how a decrease in oil supply to a country (e.g.in a war conflict) may cascade to a decrease in fuel supply to airports and flights.Another scenario that is likely to happen in the future is an increase in air traffic e.g.in Asian and African countries, which would result in a change in the infrastructure ofthe consumer and resource network.

Here we introduce and present a framework for analyzing and understanding in-terdependencies of and organizational principles of these network via a toy model.Questions we would like to investigate are: 1) under what conditions do changes in onenetwork induce changes in a dependent network? 2) Can small changes in one networkinduce large changes in another network? A toy model inspired by the aviation (con-sumer network) and the oil flow (resource network) network was designed to answerthe above questions.

We hypothesize that the structure of the intermediate layer that connects two cou-pled networks has important implications for the functioning of the network. Whenlinks going into and out of this intermediate layer are concentrated on a small numberof nodes, we anticipate that the worst case consequences of node removal increase. Toassess concentration, we utilize entropy measures for indegree and outdegree of nodesin the intermediate layer.

2 Methods

2.1 Defining an interconnected/multiplex graph

An interconnected graph can be defined as the union of graphs G = G1 ∪ G2 ∪ ... =(V1 ∪ V2 ∪ ..., E1 ∪ E2∪, ...), where Ei is a set of weighed edges representing a kind ofrelationship of vertices on set Vi. Further, a multiplex graph (V1 ≡ V2 ≡ ...) can be

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A) B) C)

Figure 1: Process of normalizing layers in a multiplex network. A) Initially we have the sameset of nodes with three different kind of edges. Each edge relationship is represented as a dif-ferent adjacency matrix Ak. B) Each edge relationship is normalized and the normalizationis represented as Ak/

∑ij A

kij. C) Finally, equation (1) is applied with w1 = w2 = w3 = 1.

represented as G = (V, 0E1 ∪E2∪, ...). For example we can have that V represent0 thecities of a country, and E1, E2, and E3 the number of people that travels across citiesby plane, train and bus, respectively.

Each graph in the interconnected/multiplex network can be represented with theadjacency matrix Ak, such that all of them have the same size and a row/columnsrepresent the same node across all the matrices. Therefore empty rows/columns needto be included for the case of interconnected networks

2.2 Normalizing the interconnected/multiplex graph

We want to normalize all the edge weights, such that we can have a single adjacencymatrix B that will represent the interconnected/multiplex graph. The elements of thismatrix are defined as:

Bij =∑k

(wk ∗

Akij∑

mnAkmn

), (1)

such that wk <= 1 quantify how important is the interconnected network (or multiplexlayer) in the graph. These values need to be assigned according to external information.For example in the case of a transportation multiplex network composed of fly, trainand bus layers, the weights can be assigned according to the total flux of users. Anotherexample can be different snapshots in time of the same graph. In this case, we mayassign the values according to the rule wt < wt+1. Figure 1 shows an small example.

2.3 Applications

This way of normalizing the graph can be used to apply any kind of metric directly to amultiplex or interconnected network. Further, we can variate the weights in Equation(1) to study how measures as centrality, modularity,etc depend on the different layersof the network.

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2.4 Toy Model

The toy model of our multiplex network consists of three layers that we can define asthe resource layer network (oil network), the consumption layer network (fly network),and the intermediate layer that connects the oil and fly network.

This three-network structure is built from the bottom-up form resource to interme-diate to consumer layers. Constraints are modeled into the networks layer by layer sothat resource carriers cannot trade resources they don’t possess with other resource car-riers, so that consumers cannot fly to destinations that they don’t obtain the requisiteresources for.

2.5 Objectives

The main reason of this toy model is that we can create synthetic data that will allowus to study the dynamics of this kind of networks with the final goal of integratingreal data. Another reason of using synthetic data first is that the real data maycontain things that are not realistic (nodes that are sinks or sources). Ideally in theconsumption network, any node can be a sink or source because that will mean thatplanes are created or destroyed from nowhere.

2.5.1 Time line

The time line for the constructing this toy model will be:

• Construct the resource layer network synthetically and test it using differentdegree distributions and random assignments for the required parameters (seebelow).

• Integrate the consumption layer network to the toy model ensuring that con-straints are met. that is free of inconsistencies (nodes that are sinks or sources).

• Study the dynamics of the toy model under different types of disturbance.

• From the real oil network data we will try to infer what is the degree distributionand random distribution of the production, consumption, etc. And we will usethis values to further study the oil network layer dynamics.

• Infer the structure of real world networks like that of oil and flight and createstylized model data to reflect this structure. Simulate network dynamics in thisstylized setting.

3 Network generation

3.1 Resource layer network

We define a resource layer network as a weighted digraph GR = (V,E) that representsthe flux of resources across different nodes, where the set V represent a set of nodesand E the set of edges. An edge i → j exist when node i exports wij > 0 resourcesto node j. A node can import and export to the same node. Further, at each time teach node i produces and consumes an amount pi(t) and ci(t) of resources, respectively.

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Finally, each year the amount of reserves for each node is updated according to thefollowing rule:

ri(t + 1) = current reserves + production− consumption + imports− exports

= ri(t) + pi(t)− ci(t) +∑j

wji −∑j

wij

3.2 Intermediate layer network

We define the intermediate (or resource-consumer) layer as edges that link resourcecarriers to consumers. The link is one way from resource carriers to consumers, andweights represent the amount of flux from resource carrier to consumer.

3.3 Consumer layer network

We define a consumer layer network as a weighted digraph GC = (V,E) that repre-sents the resource-mediated relationship between different consumers. Edges betweenconsumers are directed and require the a dependency on resources where the resourcesare not renewable or transferrable after use.

3.4 Simulation

3.4.1 Initialization

The simulation starts with a random configuration of:

• Random structure of graph GR. The graph starts with |V | = N , with a degreedistribution either from Erdos-Renyi or scale free random networks.

• Each link in the graph will have a weight randomly assigned in the range wij ∈U(wmin, wmax)

• Each node will start with random assignments for its reserve, production andconsumption at t = 0. That is we will randomly create ri(0) ∈ U(rmin, rmax),pi(0) ∈ U(pmin, pmax), ci(0) ∈ U(cmin, cmax). Further, for this part of the sim-ulation (only the resource layer), we will consider that pi(t + 1) = pi(t) andci(t + 1) = ci(t). In other words, the production and consumption of each nodeis constant during the entire simulation.

Ultimately our goal is to use a random uniform distribution for the initial assign-ments of the required quantities for the simulation. In future, the distribution ofresources will model the actual distribution of oil, so that power law or normal distri-butions may be used.

For this initial phase, we will commence with a uniform assignment of the sameamount of resources to each resource carrier in terms of production, and assume noreserves. This will help make our models more tractable. Once dynamics under thisconfiguration are understood, we can alter the “initial conditions” that inform oursimulations.

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3.4.2 Simulation step

At each simulation time t we perform the following steps:

• Node selection: A node i is selected (either randomly or by a deterministic pro-cess).

• Node deletion: The node i is deleted. This will cause that its export and importlinks to be deleted.

• Graph rewiring: The deletion of node i will cause the decrease of resources in someof the nodes. However, we only need to focus for the nodes in which rj(t+1) ≤ 0.Two possibilities exist:

– Delete some export links of node j randomly until rj(t + 1) ≥ 0

– If the last step is not possible delete node j

• Repeat last step as long as nodes with negative reserves exist.

• Re-add deleted node and its links, and commence simulation again.

For this paper, we will generate all valid networks where there are two consumersand two resource carriers. We will then generate all valid networks where there arethree

4 Hypotheses and Results

4.1 Hypotheses

Hypothesis 1. The higher the density of the resource-consumer layer, the lower theimpact of a worst case impact from any single node removal.

The intuition behind the first hypothesis regards redundancy. The more dense thatthe resource consumer layer is, the more opportunity for redundancy to consumers.This minimizes the impact of any single node’s removal.

Hypothesis 2. As the Shannon entropy increases from zero for the outdegree of theresource-consumer layer, the lower the impact of a worst case impact from any singlenode removal.

The second hypothesis relates to the concentration of outgoing resources from re-source carriers to consumers. The more that resources flow out in concentrated ways,the easier it is to knock out a node and thereby create large impacts. CalculatingShannon’s diversity index provides a measure of concentration, and higher diversityequates to lower maximum loss.

Hypothesis 3. As the Shannon entropy increases from zero for the indegree of theresource-consumer layer, the higher the impact of a worst case impact from any singlenode removal.

For the last hypothesis, concentration of indegree is another way of saying thatconsumers have redundant sources of resources to utilize. Therefore, higher Shannondiversity scores regarding indegree should translate to higher maximum impact whena single resource node is removed. Of course, consumers that get horde resource links(and also consume them) might cause large systemic issues if removed. The issues ofconsumer removal is a subject for subsequent research.

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Figure 2: All possible two by two graphs. For each graph, the bottom row holds nodes thatare resource carriers, and the top row holds consumers.

4.2 Generation of network models

Figure 2 shows the set of 108 valid graphs that are possible when each resource carrieris assigned a production value of 2 units, and all links are of unit weight. Each resourcenode is assigned a production value of n, being the number of consumer nodes. Thismeans that, independent of imports from others, a resource carrier can on its ownprovide resources to all consumers. In our simplified model, resource carriers can onlyprovide a single unit of resource to consumers.

A similar image for all 3x3 proved time prohibitive, as the image would consists of809,664 individual graphs.

4.3 Results

To test our hypotheses, we utilize simple linear regression techniques. Table 1 presentsregression tables obtained from our experiments with all 2x2 graphs. Our dependentvariable is the lowest resulting number of consumer-to-consumer links remaining afterany single resource node is removed from the network. From Table 1 Column 1, wefind evidence that supports all three hypotheses. However, there is still a possibilityof missing variable bias. This question could be alleviated by accounting for more ofour data’s variance. The inclusion of an indicator for ”singleton consumers” increasesr-squared to above 80%. A singleton consumer is one that has a degree of one onthe resource-consumer layer, while having an outdegree of 1 in the consumer-consumerlayer. Singleton consumer nodes can suffer greatly from the deletion of the singleresource node. In graphs with only 1 consumer to consumer link, this can have a

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considerable negative impact. For the 2x2 graph scenario, 80% of graphs have onlyone consumer-consumer link. 25% of graphs have both singleton consumers and onlyone consumer-consumer link.

Table 1: Maximum negative impact on consumer-to-consumer links based on r-c networkstructure in a 2-resource by 2-consumer layer

(1) (2)max impact of node removal

Shannon index forr-c outdegree 0.540∗∗∗ 0.358∗∗∗

(3.50) (4.01)

Shannon index forr-c indegree -0.787∗∗∗ -0.453∗∗∗

(-4.67) (-4.58)

r-c link density 0.425∗∗∗ 0.169∗∗∗

(5.88) (3.76)

presence of1-degree in/out consumers -0.739∗∗∗

(-14.69)

Constant -0.368∗∗ 0.513∗∗∗

(-3.18) (5.74)Observations 108 108R2 0.498 0.838

t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

To further validate our hypotheses, we conduct tests on all possible 3x3 networks.There are 809,664 graphs in this set. Table 2 presents regression models, this timeexcluding the control variable regarding singleton consumers, as singleton consumersare a collinear term with our entropy measures for this larger data set. Table 2 Column1 shows the regression model run against all graphs. R-squared here is at 32%, with allhypotheses supported in the model. After noticing that model strength improved whenrestricting data to those of higher rc network density, Columns 2 and 3 were added toTable 2, where r-c density is restricted to above 60% and above 80% respectively. Herewe also find that the the variance explained goes up to 58% and 70% with subsequentmodels, and that the impact of shannon entropy for both indegree and outdegreeincreases manyfold. This gives evidence that our hypotheses hold stronger weight insettings of higher r-c density. Why is this?

Figure 3 provides a hint of an answer. The figure shows that Shannon diversityscores of zero are found only when there are 3 or fewer c-c links. Indeed the range ofpossible Shannon scores is wider in these cases. A zero Shannon score indicates fullconcentration of outgoing links from one resource carrier to consumers. By definition

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of this experiment, one resource carrier can only link to a consumer with a unit weight.Since there a maximum of three consumer nodes, full outgoing concentration can onlyoccur in scenarios with 3 or fewer c-c links.

The figure also suggests the idea that networks that aren’t fully utilized may beharder to predict. This links to the idea that utilization tightens the ”coupling” betweeninterconnected networks, and that the dynamics of networks are more attuned to oneanother at higher utilization rates. We could imagine a similar scenario where resourcecarriers were fully utilized, so that a sudden export fluctuation in oil could cause rippleeffects through the consumer and resource layers. Such ripple might be lessened if therewas ”slack” capacity that weakened the tension of the ties linking the two networks.

Table 2: Maximum negative impact on consumer-to-consumer links based on r-c networkstructure in a 3-resource by 3-consumer layer

(1) (2) (3)max impact of node removal r-c link dens. > 60% r-c link dens. > 80%

Shannon index forr-c outdegree 0.542∗∗∗ 1.498∗∗∗ 2.240∗∗∗

(106.83) (151.45) (99.83)

Shannon index forr-c indegree -0.205∗∗∗ -0.583∗∗∗ -3.232∗∗∗

(-33.38) (-28.93) (-45.62)

r-c link density 0.375∗∗∗ 0.418∗∗∗ 0.510∗∗∗

(432.61) (312.16) (185.60)

Constant -0.578∗∗∗ -0.540∗∗∗ 1.288∗∗∗

(-90.90) (-24.50) (16.72)Observations 809664 148816 28864R2 0.328 0.578 0.702

t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

5 Future Work

Though these first steps are small and measured, this paper hopefully demonstratesthe promise of understanding the dynamics of interconnected networks in a simulatedsetting. Using this approach, we have fine tuned control over the initial conditions andstructures that affect our outcomes of interest. Future studies need to look at:

Larger networks: Due to the small number of vertices of the networks utilized inthis paper, the opportunity to look at large cascades of failures was prevented. Fu-

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Figure 3: Scatter plot of number of consumer to conusmer links vs Shannon entropy ofoutgoing degree distribution in the resource-consumer layer for 3x3 graphs

ture studies will look at how small fluctuations resonate through larger interconnectednetworks.

Real(er) data: Future studies will draw their priors from distributions found in thereal world of oil and flight networks, though they could just as easily draw priors fromthe biological world of predator and prey. Still there is a place for stylized data to geta cleaner sense of the processes at play in dynamical systems.

Tie weights: This study looked at the effects of node removal. However, ties them-selves can be lessened in terms of weight, and impacts analyzed. In addition, ties inthis study were forced to be of weight one. This stricture can be removed in futurestudies to more closely resemble resource-consumer networks in nature

Consumer nodes: This study only addressed removal of resource nodes. Futurework will address the results of removal of consumer nodes.

Unequal allocation of resources: All nodes were assigned the same production valueto allow for a more stable initial environment. Future studies will allocate initialresources from random draws from uniform, normal, and power law distributions. Theuniverse of possible interconnected models will then grow from the initial randomizedallocation of resources. This way, we can look at how small changes in initial conditionscan have broad impacts.

Tie generation: The current study generates ties in a brute force method thatlooks at all possible networks. Future studies can generate models using preferentialattachment and other means to grow networks.

Adding nodes, ties: Future studies can examine the impacts of adding nodes and

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ties to existing interconnected networks. This way, we can study questions like theimpact of growth in aviation demand in the Global South on the networks of global oiland aviation, or the impact of this change on global greenhouse emissions.

Resource types: This study focused on resources that can be consumed only once.However resource like information exists that can be reused. Also, resources like oilgenerate flights at first exposure. Other resources require multiple exposures to gener-ate link making activity (such as fads, diseases, etc). Future studies can address thesevery different dynamics.

Interconnection types: This study focused on a type of interconnection where theconsumption of a freely traded resource enables consumers to connect. There couldbe other interconnection types that bring disparate networks together. For example.financial firms might be linked by transactions that rely on a network of regulators thatact as ”gatekeepers”. Such an interconnection would have very different dynamics thanthe resource-consumer model.

6 Conclusion

This paper presents a set of tools for studying multiple interconnected networks. Uti-lizing simulation, a controlled environment can be created in which to understandmultinetwork dynamics at small and large scales. The paper uses simulation to lookat interconnected networks of resources and consumers. The paper examines whenthe removal of any single node in the resource network has the largest impact on theconsumer network. The paper finds that it’s import to focus on the network layerthat connects the the separate networks of resources and consumers. In a controlledsetting, the paper finds that increasing network density of the resource-consumer layerhas a positive impact on minimizing negative consumer impact. Increasing Shannon’sdiversity index on outgoing degree in the resource-consumer layer minimizes worst casenegative impact. Increasing Shannon’s diversity on indegree maximizes negative im-pact. The paper also finds that separate networks are more ”attuned” to one anotherunder conditions of higher utilization. There is a wide avenue of future opportuni-ties for further research in this new field of burgeoning interest in the social networksliterature.

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References

[1] Radicchi, Filippo, and Alex Arenas. “Abrupt transition in the structural formationof interconnected networks.” arXiv preprint arXiv:1307.4544 (2013).

[2] Buldyrev, Sergey V., et al. “Catastrophic cascade of failures in interdependentnetworks.”Nature 464.729: 1025-1028 (2010).

[3] Gao, Jianxi, et al. “Networks formed from interdependent networks.” NaturePhysics 8.1: 40-48 (2011).

[4] Shannon, Claude E. “A mathematical theory of communication.” The Bell SystemTechnical Journal 27: 379-423 and 623-656 (1948).

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